In quantum mechanics, the energies of cyclotron orbits of charged particles in a uniform magnetic field are quantized to discrete values, thus known as Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau.

Landau quantization contributes towards magnetic susceptibility of metals, known as Landau diamagnetism. Under strong magnetic fields, Landau quantization leads to oscillations in electronic properties of materials as a function of the applied magnetic field known as the De Haas–Van Alphen and Shubnikov–de Haas effects.

Landau quantization is a key ingredient in explanation of the integer quantum Hall effect.

Consider a system of non-interacting particles with charge q and spin S confined to an area A = L_xL_y in the x-y plane. Apply a uniform magnetic field ${\displaystyle \mathbf {B} ={\begin{pmatrix}0\\0\\B\end{pmatrix}}}$ along the z-axis. In SI units, the Hamiltonian of this system (here, the effects of spin are neglected) is $${\displaystyle {\hat {H}}={\frac {1}{2m}}\left({\hat {\mathbf {p} }}-q{\hat {\mathbf {A} }}\right)^{2}.}$$ Here, ${\textstyle {\hat {\mathbf {p} }}}$ is the canonical momentum operator and ${\textstyle {\hat {\mathbf {A} }}}$ is the operator for the electromagnetic vector potential ${\textstyle \mathbf {A} }$ (in position space ${\textstyle {\hat {\mathbf {A} }}=\mathbf {A} }$).

The vector potential is related to the magnetic field by ${\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} .}$

There is some gauge freedom in the choice of vector potential for a given magnetic field. The Hamiltonian is gauge invariant, which means that adding the gradient of a scalar field to A changes the overall phase of the wave function by an amount corresponding to the scalar field. But physical properties are not influenced by the specific choice of gauge.

From the possible solutions for A, a gauge fixing introduced by Lev Landau is often used for charged particles in a constant magnetic field.

When ${\displaystyle \mathbf {B} ={\begin{pmatrix}0\\0\\B\end{pmatrix}}}$ then ${\displaystyle \mathbf {A} ={\begin{pmatrix}0\\B\cdot x\\0\end{pmatrix}}}$ is a possible solution in the Landau gauge (not to be mixed up with the Landau ${\displaystyle R_{\xi }}$ gauge).